Searcharxiv⌕ Search

arXiv subjects

Jinjiang Li

Publications and source records attributed to Jinjiang Li.

43 records · Page 3Linked to original sources
arXiv2017

Waring-Goldbach Problem: One Square, Four Cubes and Higher Powers

Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, it is proved that, for $12\leqslant b\leqslant 35$ and for every sufficiently large odd integer $N$, the equation \begin{equation*} N=x^2+p_1^3+p_2^3+p_3^3+p_4^3+p_5^4+p_6^b \end{equation*} is solvable with $x$ being an almost-prime $\mathcal{P}_{r(b)}$ and the other variables primes, where $r(b)$ is defined in the Theorem. This result constitutes an improvement upon that of Lü and Mu.

math.NT↗
arXiv2017

A Remark on the Piatetski-Shapiro-Hua Theorem

In this paper, we prove that for any fixed $205/243<γ\leqslant1$, every sufficiently large $N$ satisfying $N\equiv 5 \pmod {24}$ can be represented as five squares of primes with one prime in $\mathcal{P}_γ$, which improves the previous result of Zhang and Zhai.

math.NT↗
arXiv2016

On Two Diophantine Inequalities Over Primes

Let $1<c<37/18,\,c\neq2$ and $N$ be a sufficiently large real number. In this paper, we prove that, for almost all $R\in(N,2N],$ the Diophantine inequality $|p_1^c+p_2^c+p_3^c-R|<\log^{-1}N$ is solvable in primes $p_1,\,p_2,\,p_3.$ Moreover, we also investigate the problem of six primes and prove that the Diophantine inequality $|p_1^c+p_2^c+p_3^c+p_4^c+p_5^c+p_6^c-N|<\log^{-1}N$ is solvable in primes $p_1,\,p_2,\,p_3,\,p_4,\,p_5,\,p_6$ for sufficiently large real number $N$.

math.NT↗
arXiv2016

On the Sum of Divisors of Mixed Powers

Let $d(n)$ denote the Dirichlet divisor function. Define \begin{equation*} \mathcal{S}_{k}(x)=\sum_{\substack{1\leqslant n_1,n_2,n_3 \leqslant x^{1/2} \\ 1\leqslant n_4\leqslant x^{1/k} }} d(n_1^2+n_2^2+n_3^2+n_4^k), \qquad 3\leqslant k\in \mathbb{N}. \end{equation*} In this paper, we establish an asymptotic formula of $\mathcal{S}_k(x)$ and prove that \begin{equation*} \mathcal{S}_k(x)=C_1(k)x^{3/2+1/k}\log x+C_2(k)x^{3/2+1/k}+O(x^{3/2+1/k-δ_k+\varepsilon}), \end{equation*} where $C_1(k),\,C_2(k)$ are two constants depending only on $k,$ with $δ_3=\frac{19}{60},\,δ_4=\frac{5}{24},\,δ_5=\frac{19}{140},\,δ_6=\frac{25}{192},\, δ_7=\frac{457}{4032},\,δ_k=\frac{1}{k+2}+\frac{1}{2k^2(k-1)}$ for $k\geqslant8.$

math.NT↗
arXiv2016

On the Seventh Power Moment of $Δ(x)$

Let $Δ(x)$ be the error term of the Dirichlet divisor problem. In this paper, we establish an asymptotic formula of the seventh-power moment of $Δ(x)$ and prove that \begin{equation*} \int_2^T Δ^7(x)\mathrm{d}x= \frac{7(5s_{7;3}(d)-3s_{7;2}(d)-s_{7;1}(d))}{2816π^7}T^{11/4}+O(T^{11/4-δ_7+\varepsilon}) \end{equation*} with $δ_7=1/336,$ which improves the previous result.

math.NT↗